Time series is statistical data that we arrange and present in a chronological order spreading over a period of time. Time series analysis is a statistical technique dealing with time series data. According to Spiegel, “A time series is a set of observations taken at specified times, usually at equal intervals.” In statistics, for time series analysis two main categories of models are popular. Let us discuss the Models of Time Series Analysis in details.

**Models of Time Series Analysis **

In time series quantitative data are arranged in the order of their occurrence and resulting statistical series. The quantitative values are usually recorded over equal time intervals such as daily, weekly, monthly, quarterly, half-yearly, yearly, or any other measure of time.

Some examples are statistics of Industrial Production in India on a monthly basis, birth-rate figures annually, the yield on ordinary shares, and weekly wholesale price of rice, etc.

**Components of Time Series**

There is a different kind of forces which influence the time series analysis. Some are continuously effective while others make themselves felt at recurring time intervals. So, our first task is to divide the data and elements into components.

A time series consists of the following four components or basic elements:

- Basic or Secular or Long-time trend;
- Seasonal variations;
- Business cycles or cyclical movement; and
- Erratic or Irregular fluctuations.

These components provide a basis for the explanation of the behavior on the past time. With their help, one can predict the behavior ahead. The major tendency of each component or constituent is largely due to causal factors.

**Mathematical Statements of Time Series**

Some time series may not be affected by all type of variations. Some of these types of variations may affect a few time series only. Hence, while analyzing the time series, these effects are isolated. In a traditional time series analysis, we assume that any given observation consists of the trend, seasonal, cyclical and irregular movements.

**Models of Time Series Analysis**

The following are the two models which we generally use for the decomposition of time series into its four components. The objective is to estimate and separate the four types of variations and to bring out the relative effect of each on the overall behavior of the time series.

(1) Additive model, and

(2) Multiplicative model

**1) Additive Model**

In the additive model, we represent a particular observation in a time series as the sum of these four components.

i.e. O = T + S + C + I

where O represents the original data, T represents the trend. S represents the seasonal variations, C represents the cyclical variations and I represents the irregular variations.

In another way, we can write Y(t) = T(t) + S(t) +C(t) + I(t)

**2) Multiplicative Model**

In this model, four components have a multiplicative relationship. So, we represent a particular observation in a time series as the product of these four components:

i.e. O = T × S × C × I

where O, T, S, C and I represents the terms as in additive model.

In another way, we can write Y(t) = T(t) × S(t) × C(t) × I(t)

This model is the most used model in the decomposition of time series. To remove any doubt between the two models, it should be made clear that in Multiplicative model S, C, and I are indices expressed as decimal percentages whereas, in Additive model S, C and I are quantitative deviations about a trend that can be expressed as seasonal, cyclical and irregular in nature.

Source: freepik.com

**Example:**

If in a multiplicative model.

T = 500, S = 1.4, C = 1.20 and I = 0.7

then O=T × S × C × I

By substituting the values we get

O = 500 × 1.4 × 1.20 × 0.7 = 588

If in additive model,

T = 500, S = 100, C = 25, I = –60

then O = 500 + 100 + 25 – 60 = 565

**Solved Question on Models of Time Series Analysis**

Q. Which model is more appropriate for time series analysis?

Solution: The assumption for the two schemes of analysis is that whereas there is no interaction among the different constituents or components under the additive scheme, such interaction is very much present in the multiplicative scheme. They do not depend on the level of the trend. With higher trends, these variations are more intensive. Though in practice the multiplicative model is the more popular, both models have their own merits. Depending on the nature of the time series analysis, they are equally acceptable.